NCERT Class 11-Math՚S: Chapter – 14 Mathematical Reasoning Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Example 12: Write the negation of each of the following disjunction:

(a) Ram is in Class X or Rahim is in Class XII.

(b) is greater than or is less than .

Answer:

(a) Let : Ram is in Class X and : Rahim is in Class XII.

Then the disjunction in is given by .

Now : Ram is not in Class X.

: Rahim is not in Class XII.

Then, using , negation of is given by

: Ram is not in Class X and Rahim is not in Class XII.

(b) Write is greater than 4, and is less than .

Then, using , negation of is given by

is not greater than and is not less than .

14.1. 11 Negation of a negation As already remarked the negation is not a connective but a modifier. It only modifies a given statement and applies only to a single simple statement. Therefore, in view of and , for a statement p, we have

Negation of negation of a statement is the statement itself. Equivalently, we write

14.1. 12 The conditional statement Recall that if p and q are any two statements, then the compound statement “if p then q” formed by joining p and q by a connective ‘if then’ is called a conditional statement or an implication and is written in symbolic form as or . Here, p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement

Remark The conditional statement can be expressed in several different ways. Some of the common expressions are:

(a) If , then

(b) if

(c) only if

(d) is sufficient for

(e) is necessary for .

Observe that the conditional statement reflects the idea that whenever it is known that is true, it will have to follow that is also true.

Example 13: Each of the following statements is also a conditional statement.

(i) If , then Rekha will get an ice-cream.

(ii) If you eat your dinner, then you will get dessert.

(iii) If John works hard, then it will rain today.

(iv) If ABC is a triangle, then .

Example 14: Express in English, the statement , where

p: it is raining today

Solution: The required conditional statement is “If it is raining today, then

14.1. 13 Contrapositive of a conditional statement: The statement “” is called the contrapositive of the statement

Example 15: Write each of the following statements in its equivalent contrapositive form:

(i) If my car is in the repair shop, then I cannot go to the market.

(ii) If Karim cannot swim to the fort, then he cannot swim across the river.

Solution: (i) Let “p: my car is in the repair shop” and “ can-not go to the market” .

Then, the given statement in symbolic form is pq. Therefore, its contrapositive is given by .

Now : My car is not in the repair shop.

and : I can go to the market

Therefore, the contrapositive of the given statement is

“If I can go to the market, then my car is not in the repair shop” .

(ii) Proceeding on the lines of the solution of (i) , the contrapositive of the statement in (ii) is

“If Karim can swim across the river, then he can swim to the fort” .

14.1. 14 Converse of a conditional statement: The conditional statement “” is called the converse of the conditional statement “

Example 16: Write the converse of the following statements

(i) If , then

(ii) If ABC is an equilateral triangle, then ABC is an isosceles triangle

Solution: (i) Let

Therefore, the converse of the statement is given by

″ If , then

(ii) Converse of the given statement is

“If ABC is an isosceles triangle, then ABC is an equilateral triangle.”